**Schedule:**

We will meet until August 29, 2019, inclusive.

Huron Classroom Thursdays 4:00 – 6.00

**About:**

- I will try to discuss a backbone of the topics presented in the first
**five**chapters of the book. Sometimes I will skip whole sections completely for the sake of time so that we can get an idea of what the topic is about. This is not so terrible since many results are familiar from other areas (i.e. several construction in Lie algebras) and we have many analogies with the known case of smooth manifolds. - I try to update this page frequently and upload the notes I use whenever I use them. Notice that my notes and my presentation
**may differ**in order or content slightly, but if you miss one lecture and see what happens in the notes you will be fine next time regardless. - I am not an expert in this topic, I am learning as well so please tell me mistakes or comments or references I could add.
- We begin at 4.10 but from 4:00 to 4.10 I talk about a “nice fact” of something wlse that I am reading, not necessarily of mathematics.

**Bibliography:**

[1] *p -adic Lie Groups*, Peter Schneider.

[2] *Arithmetic Differential Operators over the p-adic Integers*, Claire C. Ralph, Santiago R. Simanca

Mostly for comparison and review of topology:

[3] *Introduction to Topological Manifolds*, Second Edition, John M. Lee, Springer GTM 202

[4] *Introduction to Smooth Manifolds*, First Edition, John M. Lee, Springer GTM 218

**Logbook:**

Date | What did we do? | Notes |

July 4 | We discussed mostly the concept of spherically complete spaces and lemma 1.4 [1]: Section 1.1. | N/A |

July 11 | We talked about Mahler Expansion Theorem. [2]: Chapter 3, pg 23 to 29. | |

July 18 | We discussed in a different order most of the material about power series [1] Section 1.5[1] Section 1.6, discussed only the beginning. | |

July 25 | We began manifold theory and proved halfway of Theorem 8.7 of [1] (i.e. up to page 16 of the notes). [1] Section 2.7, 2.8 | |

August 1 | We saw up to page 12 of the notes. We defined tangent plane, tangent bundle and Lie Group. [1] Section 2.9, only the constructions of tangent plane and tangent bundle. [1] Section 3.13, results 13.1 to 13.5 | |

August 8 | We saw the notes in its entirety: we studied the algebraic properties of p-valuations. [1] Section 5.23, 5.25. | |

August 15 | I improvised the lecture around the topics of section 26of [1]. Of particular importance are the definition of saturated, Lemma 26.13, Proposition 26.15 and 26.16, allof [1]. | N/A |

August 22 | We started the proof of theorem 27.1 of [1]. We constructed the open compact subgroup G’. | |

August 29 | We finished the proof of theorem 27.1 of [1]. We constructed the valuation and proved its properties. |

**Notes:**

**[1] Section 1.2**: In here the*continuos dual space*is defined. We skipped this section since we use our intuition of the archimedean case and change it as we need (we do this just for the sake of time!).**[1] Section 1.3:**Deals with*convergence*. In particular it proves the concept of conditional convergence doesn’t exist in this case.**Inverse Function Theorem:**Compare the proof of the*inverse function theorem*([1], Proposition 4.3) based on [1] Lemma 4.2, with the classical case in [4], Theorem 7.6.**Point of expansion:**We did not prove*Point of expansion*([1] Corollary 5.5) but the proof is instructive.**Characteristic:**So far, we have made no distinction by field characteristic but some theorems and results require it or have a different proof. See for example*[1] Corollary 5.7, 5.8*.**Paracompactness:**To review this concept see*[3], chapter 4*. For us*[3] Lemma 4.80*and*[3] Theorem 4.81*are the important ones. Notice that part of the proof of*[1] Theorem 8.7*is exactly*[3] Lemma 4.84*. What makes the difference then in the proof is the third claim in (i) implies (ii).

**Alternative References:**

# | Reference: |

1 | Summary on non-Archimedean Valued FieldsAngel Barría Comicheo, Khodr Shamseddine Advances in Ultrametric Analysis, AMS, CONM, Vol 74 |

2 | A journey through the history of p-adic numbers Yvette Perrin Advances in Ultrametric Analysis, AMS, CONM, Vol 74 |

3 | Lectures on Lie groups over local fieldsHelge Glockner |

4 | On p-saturable groupsJon González-Sánchez |

5 | Continuous representation theory of p-adic Lie GroupsPeter Schneider |